Population Genetics using Trees. Peter Beerli Genome Sciences University of Washington Seattle WA

Transcription

1 Population Genetics using Trees Peter Beerli Genome Sciences University of Washington Seattle WA

2 Outline 1. Introduction to the basic coalescent Population models The coalescent Likelihood estimation of parameters of interest Why do we need Markov chain Monte Carlo 2. Extensions and examples

3 Population genetics can help us to find answers the PCR revolution allows us to generate lots of data from many individuals and many loci We are still interested in questions like Where are we or other species coming from? How big are populations? Are these populations species? What is the recombination rate in species x?

4 Population genetics in the age of genomics Why do we need theoretical population genetics when we can have the complete sequences of our favorite organism?

5 Basics: Wright-Fisher population model All individuals release many gametes and new individuals for the next generation are formed randomly from these.

6 Basics: Wright-Fisher population model All individuals release many gametes and new individuals for the next generation are formed randomly from these.

7 Basics: Wright-Fisher population model All individuals release many gametes and new individuals for the next generation are formed randomly from these.

8 Basics: Wright-Fisher population model All individuals release many gametes and new individuals for the next generation are formed randomly from these.

9 Basics: Wright-Fisher population model All individuals release many gametes and new individuals for the next generation are formed randomly from these.

10 Basics: Wright-Fisher population model All individuals release many gametes and new individuals for the next generation are formed randomly from these.

11 Wright-Fisher population model Population size N is constant through time. Each individual gets replaced every generation. Next generation is drawn randomly from a large gamete pool. Only genetic drift is manipulating the allele frequencies.

12 The Coalescent Sewall Wright showed that the probability that 2 gene copies come from the same gene copy in the preceding generation is Prob (two genes share a parent) = 1 2N

13 The Coalescent Present Past In every generation, there is a chance of 1/2N to coalesce. Following the sampled lineages through generations backwards in time we realize that it follows a geometric distribution with E(u) = 2N [the expectation of the time of coalescence u of two tips is 2N]

14 The Coalescent JFC Kingman generalized this for k gene copies. Prob (k copies are reduced to k 1 copies) = k(k 1) 4N

15 Kingman s n-coalescent Present Past

16 Kingman s n-coalescent Present The expectation for the time interval u k is E(u k ) = 4N k(k 1) u 4 u 3 u 2 Past p(g N) = i exp( u i k(k 1) 4N ) 1 2N

17 BUT, what s this good for????????????????????

18 Naively we could estimate: 1.Time of the most recent common ancestor For a given population size we can calculate the time of the most recent common ancestor [MRCA]. 1. Get a TRUE genealogy (topology and branch lengths) from an infallible oracle. 2. Get the population size from the same oracle. 3. Calculate the time of the MRCA by summing over all time intervals.

19 1. Time of the most recent common ancestor [Shortcut] 1. Get the population size from another oracle 2. Use the expectation for your data type to get an estimate of the time of the MRCA The expectation for the time of the MRCA is E(u) = 4N E(u) = 2N E(u) = N for diploid organisms for haploid organisms for maternally transmitted mtdna, paternally transmitted Y-chromosome [assumption: sex-ratio is 1:1]

20 2. Calculate the size of the population 1. We get THE genealogy from our oracle 2. We know that we can calculate p(genealogy N)

21 2. Calculate the size of the population 1. We get THE genealogy from our oracle 2. We remember the probability calculation p(g N) = p(u 1 N, k) 1 2N p(u 2 N, k 1) 1 2N...

22 2. Calculate the size of the population 1. We get THE genealogy from our oracle 2. We remember the probability calculation p(genealogy N) = T j e u k j (k j 1) j 4N 1 2N

23 2. Calculate the size of the population

24 2. Calculate the size of the population

25 2. Calculate the size of the population

26 2. Calculate the size of the population

27 2. Calculate the size of the population

28 2. Calculate the size of the population

29 2. Calculate the size of the population

30 2. Calculate the size of the population

31 2. Calculate the size of the population

32 2. Calculate the size of the population

33 2. Calculate the size of the population N = 2270 N = 12286

34 Problems with these very naive approaches We assume we know the TRUE genealogy: topology and branch length.

35 Variability of the coalescent 10 coalescent trees generated with the same population size, N = 10, 000

36 Variability of mutations

37 How many samples do we need?

38 Summary of the basic Coalescent Mathematically tractable way to calculate probabilities of genealogies in a population. The coalescent is a very noisy distribution of times on a tree. Variability because of mutation increases the uncertainty of these times. The population size is correlated with the depth of the tree. Estimations of population size or the time of the MRCA from a single tree are very error-prone.

39 Parameter estimation using maximum likelihood Mutation model: Nucleotide mutation model,... Population genetics model: the Coalescent

40 Parameter estimation using maximum likelihood Mutation model: Nucleotide mutation model,... Population genetics model: the Coalescent Prob (N, µ data)

41 Parameter estimation using maximum likelihood Mutation model: Nucleotide mutation model,... Population genetics model: the Coalescent Prob (N, µ data) = Prob (data N, µ) Prob (N, µ) Prob (data)

42 Parameter estimation using maximum likelihood Mutation model: Nucleotide mutation model,... Population genetics model: the Coalescent Prob (N, µ data) = Prob (data N, µ) Prob (N, µ) Prob (data) L(N, µ) = Prob (data N, µ) = c Prob (N, µ data)

43 Parameter estimation using maximum likelihood L(N, µ) = Prob (data N, µ)

44 Parameter estimation using maximum likelihood L(N, µ) = Prob (data N, µ) = G p(g N, µ) Prob (data G, µ)

45 Parameter estimation using maximum likelihood L(N, µ) = Prob (data N, µ) = G p(g N, µ) Prob (data G, µ) We cannot observe the mutation events. Instead of estimating N and µ we estimate the product Θ = 4Nµ and scale G with µ L(Θ) = G p(g Θ) Prob (data G )

46 Parameter estimation using maximum likelihood L(N, µ) = Prob (data N, µ) = G p(g N, µ) Prob (data G, µ) We cannot observe the mutation events. Instead of estimating N, and µ we estimate the product Θ = 4Nµ and scale G with µ L(Θ) = G p(g Θ) Prob (data G ) Problem: We need to integrate over all genealogies: all different labelled histories, all different branchlengths

47 Can we calculate this sum over all genealogies? We need to integrate over all genealogies: different branchlengths all different topologies, all Tips Topologies

48 A solution:markov chain Monte Carlo Metropolis recipe 0. first state 1. perturb old state and calculate probability of new state 2. test if new state is better than old state: accept if ratio of new and old is larger than a random number between 0 and move to new state if accepted otherwise stay at old state 4. go to 1

49 How do we change a genealogy A z B C D j k 1 2

50 Markov chain Monte Carlo L(G 1 Θ) 1

51 Markov chain Monte Carlo create a new tree 1

52 Markov chain Monte Carlo 1 2 L(G 2 Θ) Evaluate r < p(g 2 Θ)P(D G 2 )P(G 1 G 2 ) p(g 1 Θ)P(D G 1 )P(G 2 G 1 ) luckily reduces most often to: r < P(D G 2) P(D G 1 )

53 Markov chain Monte Carlo Store G 1 Make another change to the tree 1 2

54 Markov chain Monte Carlo 3 L(G 3 Θ) r < P(D G 3) P(D G 2 ) 1 2

55 Markov chain Monte Carlo 3 Store G 2 Make another change to the tree 1 2

56 Markov chain Monte Carlo 3 1 2

57 Markov chain Monte Carlo

58 Markov chain Monte Carlo

59 MCMC walk I

60 MCMC walk result Probability Tree space Tree space

61 MCMC walk result Probability Tree space Tree space

62 Improving our MCMC walker: MCMCMC or MC 3 Metropolis Coupled Markov chain Monte Carlo Run several independent parallel chains: each has a different temperature After some sampling of genealogies, swap the genealogies of a pair of chains if the ratio between probabilities in the cold and the hot chain is larger than a random number drawn between 0 and 1.

63 MCMC walk II

64 better MCMC walk result